Two Column Proofs
PROOF Geometry
Finding & Describing Patterns
• Geometry, like much of mathematics and
science, developed when people began
recognizing and describing patterns.
• Inductive reasoning is used to find and
describe patterns.
Using Inductive Reasoning
•
•
Inductive reasoning was used to discover
many of the theorems and postulates we
use today
You can use inductive reasoning to make
your own conjectures when you are
given a situation and you need to find a
conclusion
Steps involved in using
Inductive Reasoning
1. Look for a Pattern: Look at several
examples. Use diagrams and tables to
help discover a pattern.
2. Make a Conjecture. Use the example to
make a general conjecture.
A conjecture is an unproven statement
that is based on observations.
3. Discuss the conjecture with others.
Modify the conjecture, if necessary.
EXAMPLE: Making a
Conjecture
• Conjecture:
The sum of the two positive even integers
is _____________.
• How to proceed:
List some specific examples and look for a
pattern.
ExAMPLE: Making a
Conjecture
Some evens added:
2+2 = 4
2+4 = 6
4+6 = 10
6+8 = 14
EVEN
To show that a conjecture is
True
• We need to prove it for all cases.
• A proof like this will typically involve algebra,
geometry definitions, postulate and theorems
and deductive reasoning.
Let the first integer be 2n and the second 2m
Sum = 2n + 2m
=2 (n + m)
always even
To show that a conjecture is
false
• Show the conjecture is false by finding a
counterexample.
Conjecture: For all real numbers x, the
expressions x2 is greater than or equal to x.
Finding a counterexample Solution
Conjecture: For all real numbers x, the
expressions x2 is greater than or equal to x.
• The conjecture is false. Here is a
counterexample: (0.5)2 = 0.25, and 0.25 is NOT
greater than or equal to 0.5. In fact, any number
between 0 and 1 is a counterexample.
Conditional Statement
We often write conjectures as conditional statements
Definition: A conditional statement is a statement that
can be written in if-then form.
“If _____________, then
______________.”
Example: If your feet smell and your nose runs,
then you're built upside down.
Conditional Statements have two parts:
The hypothesis is the part of a conditional statement
that follows “if” (when written in if-then form.)
-The hypothesis is the given information, or the
condition.
The conclusion is the part of an if-then statement
that follows “then” (when written in if-then form.)
-The conclusion is the result of the given
information.
Every theorem can be written as a conditional
statement.
Using the Laws of Logic
• Definition:
• Deductive reasoning uses postulates,
definitions, and theorems in a logical order to
write a logical argument.
• The logical argument is called a proof
This differs from inductive reasoning, in which
previous examples and patterns are used to
form a conjecture.
Comparison of Inductive and
Deductive Reasoning
Comparison of Inductive and
Deductive Reasoning
Types of proof
• A paragraph proof
Most commonly used in upper-level mathematics
Types of proof
• A two column proof:
Most commonly used in high school.
Has numbered statements and reasons that show
the logical order of an argument.
Types of proof
• A flow chart proof
Justification in Proofs using
Properties of equality
Definition of …
• Used when using a definition of defined term to justify
statements
• Examples:
• Congruent to equal:
AB ≅ CD
AB = CD
Given
Def. of congruent segments
• Equal to congruent:
AB = CD
AB ≅ CD
Reason
Def. of congruent segments
Definition of …
CONGRUENT TO EQUAL ANGLES
Given
Definition of congruent angles
EQUAL TO CONGRUENT ANGLES
Given
Definition of congruent angles
Definition of … Example
1 and 2 are complementary
m1 + m2 = 90
Given
Def. of complementary angles
Two Column Proofs
• We will start with simple, short, easy proofs.
• With the short proofs I will show you how to
do a Formal Proof which will include writing
down each logical step – even if it seems
obvious.
• Once the proofs get more complex we will
write more Compact Proofs so we don’t get
bogged down in notation.
• Although the proofs are called compact I will
still require accurate use of logic, definitions,
postulates, theorems and the two-column
format.
Two Column Proofs
• We will use the following method:
• Create two columns, label the left Statements and the right
Reasons.
• Start with a given and if possible make some conclusions
before using another given.
• Number each line and providing a reason for each statement.
• Sometimes your reasons will include references to previous
line numbers.
• When you have finished the proof, write Q.E.D. (quod erat
demonstrandum, which is Latin for "which was to be shown")
or make a filled-in square (a "bullet") at the end of the proof.
Two Column Proof:
Algebra Example
• If 5x – 18 = 3x +2, prove that x= 10
1. 5x – 18 = 3x + 2
1. Given
2. 2x – 18 = 2
2. Subtraction prop. of
eq.
3. Addition prop. Of
eq.
4. Division prop. Of eq.
3. 2x = 20
4. x = 10
Q.E.D.
Homework
• Beginning Proofs Worksheets